Computation of rays in an inhomogeneous transverselly isotropic medium with a non-vertical axis of symmetry

1983 ◽  
Vol 27 (2) ◽  
pp. 114-121 ◽  
Author(s):  
Jiří Jech ◽  
K. Pěč
Keyword(s):  
Isotropic Medium ◽  
Vertical Axis ◽  
Axis Of Symmetry

Effects of anisotropy on time-depth relation in transversely isotropic medium with a vertical axis of symmetry

2010 ◽  
Vol 55 (21) ◽  
pp. 2243-2251 ◽  
Author(s):  
Lin Fa ◽  
John P. Castagna ◽  
ZhengWen Zeng ◽  
Ray L. Brown ◽  
MeiShan Zhao
Keyword(s):  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Transversely Isotropic Medium ◽  
Axis Of Symmetry

scholarly journals Theoretical Gravity Anomalies of Glaciers Having Parabolic Cross-Sections

1964 ◽  
Vol 5 (38) ◽  
pp. 255-257 ◽  
Author(s):  
Charles E. Corbató
Keyword(s):  
Cross Sections ◽  
Lower Boundary ◽  
Gravity Anomalies ◽  
Vertical Axis ◽  
Two Dimensional ◽  
Axis Of Symmetry ◽  
Upper Boundary

AbstractEquations and a graph are presented for calculating gravity anomalies on a two-dimensional glacier model having a horizontal upper boundary and a lower boundary which is a parabola with a vertical axis of symmetry.

scholarly journals An efficient wavefield inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R195-R206 ◽  
Author(s):  
Chao Song ◽  
Tariq Alkhalifah
Keyword(s):  
High Resolution ◽  
Optimization Problem ◽  
Source Function ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Highly Nonlinear ◽  
Isotropic Media ◽  
Axis Of Symmetry

Conventional full-waveform inversion (FWI) aims at retrieving a high-resolution velocity model directly from the wavefields measured at the sensor locations resulting in a highly nonlinear optimization problem. Due to the high nonlinearity of FWI (manifested in one form in the cycle-skipping problem), it is easy to fall into local minima. Considering that the earth is truly anisotropic, a multiparameter inversion imposes additional challenges in exacerbating the null-space problem and the parameter trade-off issue. We have formulated an optimization problem to reconstruct the wavefield in an efficient matter with background models by using an enhanced source function (which includes secondary sources) in combination with fitting the data. In this two-term optimization problem to fit the wavefield to the data and to the background wave equation, the inversion for the wavefield is linear. Because we keep the modeling operator stationary within each frequency, we only need one matrix inversion per frequency. The inversion for the anisotropic parameters is handled in a separate optimization using the wavefield and the enhanced source function. Because the velocity is the dominant parameter controlling the wave propagation, it is updated first. Thus, this reduces undesired updates for anisotropic parameters due to the velocity update leakage. We find the effectiveness of this approach in reducing parameter trade-off with a distinct Gaussian anomaly model. We find that in using the parameterization [Formula: see text], and [Formula: see text] to describe the transversely isotropic media with a vertical axis of symmetry model in the inversion, we end up with high resolution and minimal trade-off compared to conventional parameterizations for the anisotropic Marmousi model. Application on 2D real data also indicates the validity of our method.

scholarly journals Frequency domain multiparameter acoustic inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. C1-C14 ◽  
Author(s):  
Ramzi Djebbi ◽  
Tariq Alkhalifah
Keyword(s):  
Frequency Domain ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
The Other ◽  
Sensitivity Analyses ◽  
Data Set ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry

Multiparameter full-waveform inversion for transversely isotropic media with a vertical axis of symmetry (VTI) suffers from the trade-off between the parameters. The trade-off results in the leakage of one parameter’s update into the other. It affects the accuracy and convergence of the inversion. The sensitivity analyses suggested a parameterization using the horizontal velocity [Formula: see text], Thomsen’s parameter [Formula: see text], and the anelliptic parameter [Formula: see text] to reduce the trade-off for surface recorded seismic data. We aim to invert for this parameterization using the scattering integral (SI) method. The available Born sensitivity kernels, within this approach, can be used to calculate additional inversion information. We mainly compute the diagonal of the approximate Hessian, used as a conjugate-gradient preconditioner, and the gradients’ step lengths. We consider modeling in the frequency domain. The large computational cost of the SI method can be avoided with direct Helmholtz equation solvers. We applied our method to the VTI Marmousi II model for various inversion strategies. We found that we can invert the [Formula: see text] accurately. For the [Formula: see text] parameter, only the short wavelengths are well-recovered. On the other hand, the [Formula: see text] parameter impact is weak on the inversion results and can be fixed. However, a good background [Formula: see text], with accurate long wavelengths, is needed to correctly invert for [Formula: see text]. Furthermore, we invert a real data set acquired by CGG from offshore Australia. We simultaneously invert all three parameters using our inversion approach. The velocity model is improved, and additional layers are recovered. We confirm the accuracy of the results by comparing them with well-log information, as well as looking at the data and angle gathers.

The effect of transverse isotropy on isotropic traveltime inversion of vertical seismic profile data—A modeling study

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1235-1241 ◽  
Author(s):  
Jan Douma
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Seismic Profile ◽  
Isotropic Layer ◽  
Low Velocity ◽  
Traveltime Inversion ◽  
Axis Of Symmetry ◽  
Transversely Isotropic Layer

Traveltime inversion of multioffset VSP data can be used to determine the depths of the interfaces in layered media. Many inversion schemes, however, assume isotropy and consequently may introduce erroneous structures for anisotropic media. Synthetic traveltime data are computed for layered anisotropic media and inverted assuming isotropic layers. Only the interfaces between these layers are inverted. For a medium consisting of a horizontal isotropic low‐velocity layer on top of a transversely isotropic layer with a horizontal axis of symmetry (e.g., anisotropy due to aligned vertical cracks), 2-D isotropic inversion results in an anticline. For a given axis of symmetry the form of this anticline depends on the azimuth of the source‐borehole direction. The inversion result is a syncline (in 3-D a “bowl” structure), regardless of the azimuth of the source‐borehole direction for a vertical axis of symmetry of the transversely isotropic layer (e.g., anisotropy due to horizontal bedding).

scholarly journals Effect of rotation on wave propagation in a transversely isotropic medium

2001 ◽  
Vol 7 (2) ◽  
pp. 147-154 ◽  
Author(s):  
F. Ahmad ◽  
A. Khan
Keyword(s):  
Wave Propagation ◽  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Slow Waves ◽  
Fast Wave ◽  
Transversely Isotropic Medium ◽  
High Frequencies ◽  
Unbounded Medium ◽  
Axis Of Symmetry ◽  
Effects Of Rotation

Wave propagation in a transversely isotropic unbounded medium rotating about its axis of symmetry is studied. For propagation at high frequencies, effects of rotation are negligible but for a frequency which is much smaller than the frequency of rotation, there is a fast wave and two very slow waves. When the two frequencies are equal, the speed of a wave becomes unbounded.

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